Average Error: 29.8 → 5.1
Time: 19.7s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -8.829291748098453965699783086839461538272 \cdot 10^{-15}:\\ \;\;\;\;\frac{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(x \cdot \left(\frac{1}{2} \cdot {a}^{2} + \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot x\right)\right) \cdot x\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -8.829291748098453965699783086839461538272 \cdot 10^{-15}:\\
\;\;\;\;\frac{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}\\

\mathbf{else}:\\
\;\;\;\;a \cdot x + \left(x \cdot \left(\frac{1}{2} \cdot {a}^{2} + \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot x\right)\right) \cdot x\\

\end{array}
double f(double a, double x) {
        double r49198 = a;
        double r49199 = x;
        double r49200 = r49198 * r49199;
        double r49201 = exp(r49200);
        double r49202 = 1.0;
        double r49203 = r49201 - r49202;
        return r49203;
}
↓
double f(double a, double x) {
        double r49204 = a;
        double r49205 = x;
        double r49206 = r49204 * r49205;
        double r49207 = -8.829291748098454e-15;
        bool r49208 = r49206 <= r49207;
        double r49209 = 2.0;
        double r49210 = r49209 * r49206;
        double r49211 = exp(r49210);
        double r49212 = 1.0;
        double r49213 = r49212 * r49212;
        double r49214 = r49211 - r49213;
        double r49215 = exp(r49206);
        double r49216 = r49215 + r49212;
        double r49217 = r49214 / r49216;
        double r49218 = 0.5;
        double r49219 = pow(r49204, r49209);
        double r49220 = r49218 * r49219;
        double r49221 = 0.16666666666666666;
        double r49222 = 3.0;
        double r49223 = pow(r49204, r49222);
        double r49224 = r49221 * r49223;
        double r49225 = r49224 * r49205;
        double r49226 = r49220 + r49225;
        double r49227 = r49205 * r49226;
        double r49228 = r49227 * r49205;
        double r49229 = r49206 + r49228;
        double r49230 = r49208 ? r49217 : r49229;
        return r49230;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.8
Target0.2
Herbie5.1
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -8.829291748098454e-15

    1. Initial program 1.1

      \[e^{a \cdot x} - 1\]
    2. Using strategy rm

    3. Applied add-cbrt-cube1.1

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(e^{a \cdot x} - 1\right) \cdot \left(e^{a \cdot x} - 1\right)\right) \cdot \left(e^{a \cdot x} - 1\right)}}\]

    4. Simplified1.1

      \[\leadsto \sqrt[3]{\color{blue}{{\left(e^{a \cdot x} - 1\right)}^{3}}}\]
    5. Using strategy rm

    6. Applied flip--1.1

      \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}{e^{a \cdot x} + 1}\right)}}^{3}}\]

    7. Applied cube-div1.1

      \[\leadsto \sqrt[3]{\color{blue}{\frac{{\left(e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1\right)}^{3}}{{\left(e^{a \cdot x} + 1\right)}^{3}}}}\]

    8. Applied cbrt-div1.1

      \[\leadsto \color{blue}{\frac{\sqrt[3]{{\left(e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1\right)}^{3}}}{\sqrt[3]{{\left(e^{a \cdot x} + 1\right)}^{3}}}}\]

    9. Simplified1.0

      \[\leadsto \frac{\color{blue}{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}}{\sqrt[3]{{\left(e^{a \cdot x} + 1\right)}^{3}}}\]

    10. Simplified1.0

      \[\leadsto \frac{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}{\color{blue}{e^{a \cdot x} + 1}}\]

    if -8.829291748098454e-15 < (* a x)

    1. Initial program 44.8

      \[e^{a \cdot x} - 1\]

    2. Taylor expanded around 0 14.1

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right) + a \cdot x\right)}\]

    3. Simplified10.7

      \[\leadsto \color{blue}{a \cdot x + {x}^{2} \cdot \left(\frac{1}{2} \cdot {a}^{2} + \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot x\right)}\]
    4. Using strategy rm

    5. Applied sqr-pow10.7

      \[\leadsto a \cdot x + \color{blue}{\left({x}^{\left(\frac{2}{2}\right)} \cdot {x}^{\left(\frac{2}{2}\right)}\right)} \cdot \left(\frac{1}{2} \cdot {a}^{2} + \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot x\right)\]

    6. Applied associate-*l*7.2

      \[\leadsto a \cdot x + \color{blue}{{x}^{\left(\frac{2}{2}\right)} \cdot \left({x}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{1}{2} \cdot {a}^{2} + \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot x\right)\right)}\]

    7. Simplified7.2

      \[\leadsto a \cdot x + {x}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot {a}^{2} + \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot x\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification5.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -8.829291748098453965699783086839461538272 \cdot 10^{-15}:\\ \;\;\;\;\frac{e^{2 \cdot \left(a \cdot x\right)} - 1 \cdot 1}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(x \cdot \left(\frac{1}{2} \cdot {a}^{2} + \left(\frac{1}{6} \cdot {a}^{3}\right) \cdot x\right)\right) \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.10000000000000001) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))